Theorem infdifsn 9665

Description: Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
infdifsn	⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)

Proof of Theorem infdifsn

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 8994	. . . 4 ⊢ (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1→𝐴)
2	1	adantr 486	. . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃𝑓 𝑓:ω–1-1→𝐴)
3		reldom 8987	. . . . . . 7 ⊢ Rel ≼
4	3	brrelex2i 5708	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
5	4	ad2antrr 739	. . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → 𝐴 ∈ V)
6		simplr 781	. . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → 𝐵 ∈ 𝐴)
7		f1f 6783	. . . . . . 7 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω⟶𝐴)
8	7	adantl 487	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → 𝑓:ω⟶𝐴)
9		peano1 7904	. . . . . 6 ⊢ ∅ ∈ ω
10		ffvelcdm 7086	. . . . . 6 ⊢ ((𝑓:ω⟶𝐴 ∧ ∅ ∈ ω) → (𝑓‘∅) ∈ 𝐴)
11	8, 9, 10	sylancl 598	. . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝑓‘∅) ∈ 𝐴)
12		difsnen 9086	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐴 ∧ (𝑓‘∅) ∈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}))
13	5, 6, 11, 12	syl3anc 1398	. . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}))
14		vex 3454	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
15		f1f1orn 6841	. . . . . . . . . . 11 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω–1-1-onto→ran 𝑓)
16	15	adantl 487	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → 𝑓:ω–1-1-onto→ran 𝑓)
17		f1oen3g 9001	. . . . . . . . . 10 ⊢ ((𝑓 ∈ V ∧ 𝑓:ω–1-1-onto→ran 𝑓) → ω ≈ ran 𝑓)
18	14, 16, 17	sylancr 599	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ω ≈ ran 𝑓)
19	18	ensymd 9040	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ran 𝑓 ≈ ω)
20	3	brrelex1i 5707	. . . . . . . . . . 11 ⊢ (ω ≼ 𝐴 → ω ∈ V)
21	20	ad2antrr 739	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ω ∈ V)
22		limom 7897	. . . . . . . . . . 11 ⊢ Lim ω
23	22	limenpsi 9179	. . . . . . . . . 10 ⊢ (ω ∈ V → ω ≈ (ω ∖ {∅}))
24	21, 23	syl 18	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ω ≈ (ω ∖ {∅}))
25	14	resex 6020	. . . . . . . . . . 11 ⊢ (𝑓 ↾ (ω ∖ {∅})) ∈ V
26		simpr 490	. . . . . . . . . . . 12 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → 𝑓:ω–1-1→𝐴)
27		difss 4071	. . . . . . . . . . . 12 ⊢ (ω ∖ {∅}) ⊆ ω
28		f1ores 6844	. . . . . . . . . . . 12 ⊢ ((𝑓:ω–1-1→𝐴 ∧ (ω ∖ {∅}) ⊆ ω) → (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅})))
29	26, 27, 28	sylancl 598	. . . . . . . . . . 11 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅})))
30		f1oen3g 9001	. . . . . . . . . . 11 ⊢ (((𝑓 ↾ (ω ∖ {∅})) ∈ V ∧ (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅}))) → (ω ∖ {∅}) ≈ (𝑓 “ (ω ∖ {∅})))
31	25, 29, 30	sylancr 599	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (ω ∖ {∅}) ≈ (𝑓 “ (ω ∖ {∅})))
32		f1orn 6840	. . . . . . . . . . . . 13 ⊢ (𝑓:ω–1-1-onto→ran 𝑓 ↔ (𝑓 Fn ω ∧ Fun ◡𝑓))
33	32	simprbi 503	. . . . . . . . . . . 12 ⊢ (𝑓:ω–1-1-onto→ran 𝑓 → Fun ◡𝑓)
34		imadif 6629	. . . . . . . . . . . 12 ⊢ (Fun ◡𝑓 → (𝑓 “ (ω ∖ {∅})) = ((𝑓 “ ω) ∖ (𝑓 “ {∅})))
35	16, 33, 34	3syl 19	. . . . . . . . . . 11 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ (ω ∖ {∅})) = ((𝑓 “ ω) ∖ (𝑓 “ {∅})))
36		f1fn 6784	. . . . . . . . . . . . . 14 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓 Fn ω)
37	36	adantl 487	. . . . . . . . . . . . 13 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → 𝑓 Fn ω)
38		fnima 6674	. . . . . . . . . . . . 13 ⊢ (𝑓 Fn ω → (𝑓 “ ω) = ran 𝑓)
39	37, 38	syl 18	. . . . . . . . . . . 12 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ ω) = ran 𝑓)
40		fnsnfv 6969	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn ω ∧ ∅ ∈ ω) → {(𝑓‘∅)} = (𝑓 “ {∅}))
41	37, 9, 40	sylancl 598	. . . . . . . . . . . . 13 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → {(𝑓‘∅)} = (𝑓 “ {∅}))
42	41	eqcomd 2766	. . . . . . . . . . . 12 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ {∅}) = {(𝑓‘∅)})
43	39, 42	difeq12d 4063	. . . . . . . . . . 11 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ((𝑓 “ ω) ∖ (𝑓 “ {∅})) = (ran 𝑓 ∖ {(𝑓‘∅)}))
44	35, 43	eqtrd 2795	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ (ω ∖ {∅})) = (ran 𝑓 ∖ {(𝑓‘∅)}))
45	31, 44	breqtrd 5130	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (ω ∖ {∅}) ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
46		entr 9041	. . . . . . . . 9 ⊢ ((ω ≈ (ω ∖ {∅}) ∧ (ω ∖ {∅}) ≈ (ran 𝑓 ∖ {(𝑓‘∅)})) → ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
47	24, 45, 46	syl2anc 596	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
48		entr 9041	. . . . . . . 8 ⊢ ((ran 𝑓 ≈ ω ∧ ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)})) → ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
49	19, 47, 48	syl2anc 596	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
50		difexg 5291	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐴 ∖ ran 𝑓) ∈ V)
51		enrefg 9019	. . . . . . . 8 ⊢ ((𝐴 ∖ ran 𝑓) ∈ V → (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓))
52	5, 50, 51	3syl 19	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓))
53		disjdif 4425	. . . . . . . 8 ⊢ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅
54	53	a1i 11	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅)
55		difss 4071	. . . . . . . . . 10 ⊢ (ran 𝑓 ∖ {(𝑓‘∅)}) ⊆ ran 𝑓
56		ssrin 4137	. . . . . . . . . 10 ⊢ ((ran 𝑓 ∖ {(𝑓‘∅)}) ⊆ ran 𝑓 → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)))
57	55, 56	ax-mp 5	. . . . . . . . 9 ⊢ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓))
58		sseq0 4353	. . . . . . . . 9 ⊢ ((((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)
59	57, 53, 58	mp2an 705	. . . . . . . 8 ⊢ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅
60	59	a1i 11	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)
61		unen 9081	. . . . . . 7 ⊢ (((ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}) ∧ (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓)) ∧ ((ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅ ∧ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) ≈ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)))
62	49, 52, 54, 60, 61	syl22anc 852	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) ≈ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)))
63	8	frnd 6723	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ran 𝑓 ⊆ 𝐴)
64		undif 4437	. . . . . . 7 ⊢ (ran 𝑓 ⊆ 𝐴 ↔ (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) = 𝐴)
65	63, 64	sylib 221	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) = 𝐴)
66		uncom 4156	. . . . . . 7 ⊢ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)) = ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)}))
67		eldifn 4067	. . . . . . . . . . 11 ⊢ ((𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓) → ¬ (𝑓‘∅) ∈ ran 𝑓)
68		fnfvelrn 7085	. . . . . . . . . . . 12 ⊢ ((𝑓 Fn ω ∧ ∅ ∈ ω) → (𝑓‘∅) ∈ ran 𝑓)
69	37, 9, 68	sylancl 598	. . . . . . . . . . 11 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝑓‘∅) ∈ ran 𝑓)
70	67, 69	nsyl3 139	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ¬ (𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓))
71		disjsn 4668	. . . . . . . . . 10 ⊢ (((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅ ↔ ¬ (𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓))
72	70, 71	sylibr 237	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅)
73		undif4 4419	. . . . . . . . 9 ⊢ (((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅ → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}))
74	72, 73	syl 18	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}))
75		uncom 4156	. . . . . . . . . 10 ⊢ ((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) = (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓))
76	75, 65	eqtrid 2807	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) = 𝐴)
77	76	difeq1d 4061	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}) = (𝐴 ∖ {(𝑓‘∅)}))
78	74, 77	eqtrd 2795	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (𝐴 ∖ {(𝑓‘∅)}))
79	66, 78	eqtrid 2807	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)) = (𝐴 ∖ {(𝑓‘∅)}))
80	62, 65, 79	3brtr3d 5135	. . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → 𝐴 ≈ (𝐴 ∖ {(𝑓‘∅)}))
81	80	ensymd 9040	. . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝐴 ∖ {(𝑓‘∅)}) ≈ 𝐴)
82		entr 9041	. . . 4 ⊢ (((𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}) ∧ (𝐴 ∖ {(𝑓‘∅)}) ≈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
83	13, 81, 82	syl2anc 596	. . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
84	2, 83	exlimddv 1967	. 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
85		difsn 4757	. . . 4 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∖ {𝐵}) = 𝐴)
86	85	adantl 487	. . 3 ⊢ ((ω ≼ 𝐴 ∧ ¬ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) = 𝐴)
87		enrefg 9019	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ≈ 𝐴)
88	4, 87	syl 18	. . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ 𝐴)
89	88	adantr 486	. . 3 ⊢ ((ω ≼ 𝐴 ∧ ¬ 𝐵 ∈ 𝐴) → 𝐴 ≈ 𝐴)
90	86, 89	eqbrtrd 5126	. 2 ⊢ ((ω ≼ 𝐴 ∧ ¬ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
91	84, 90	pm2.61dan 825	1 ⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3896   ∖ cdif 4044   ∩ cin 4088   ∪ cun 4148  ∅c0 4277  {csn 4584   class class class wbr 5102  ^◡ccnv 5650  ran crn 5652   ↾ cres 5653   “ cima 5654  Fun wfun 6537   Fn wfn 6538  ⟶wf 6539  –1-1→wf1 6540  –1-1-onto→wf1o 6542  ‘cfv 6543  ωcom 7881   ≈ cen 8978   ≼ cdom 8979

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7882  df-er 8725  df-en 8982  df-dom 8983

This theorem is used by:  infdiffi  9666  infdju1  10275  infpss  10301  soinfdom  35774